Computations with Witt vectors of length $3$

Finotti, Luís R. A.

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 2, pp. 417-454.

Dans cet article, nous décrivons comment effectuer des calculs avec les vecteurs de Witt de longueur $3$ d’une manière efficace et donnons une formule qui permet de calculer directement la troisième coordonnée de la transformée de Greenberg d’un polynôme. Nous appliquons ces résultats afin d’obtenir des renseignements sur la troisième coordonnée de l’invariant $j$ du relèvement canonique en fonction de l’invariant $j$ de la courbe elliptique ordinaire en caractéristique $p$.

In this paper we describe how to perform computations with Witt vectors of length $3$ in an efficient way and give a formula that allows us to compute the third coordinate of the Greenberg transform of a polynomial directly. We apply these results to obtain information on the third coordinate of the $j$-invariant of the canonical lifting as a function on the $j$-invariant of the ordinary elliptic curve in characteristic $p$.

Classification : 11G20, 11Y16
Mots clés : Witt vectors, elliptic curves, canonical lifting, pseudo-canonical lifting, modular polynomial

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